chore: Update papers repository structure.

Signed-off-by: Amlal El Mahrouss <amlal@nekernel.org>

4 files changed, 151 insertions, 7 deletions

diff --git a/source/README.md b/source/README.md
index 042ae8f..7d1f81f 100644
--- a/source/README.md
+++ b/source/README.md
@@ -1,14 +1,19 @@
-# NeKernel.org Working Groups.
+# NeKernel.org Working Groups/DN Groups.
 
 #### This README indexes the different working groups and their purpose.
 
 ## Working Groups:
 
-### Here are the following active working groups:
+### Active Working Groups:
 
-- WG01: `PL/OS Group`.
-- WG02: `Systems Group`.
-- WG03: `Nectar Group`.
-- WG04-SPACE-RF: `Distributed Systems Group`.
-- WG05: `Execution Theory Group`.
+- WG01: `The PL/OS Group`.
+- WG02: `The Systems Group`.
+- WG03: `The Nectar PL Group`.
+- WG04-SPACE-RF: `The Distributed Systems Group`.
+- WG05: `The Execution Theory Group`.
+
+## Active DN Groups:
+
+- WG001: `The Mathematics of Execution.`
+- WG002: `The C Index Property.`
 
diff --git a/source/dn001.05/paper.tex b/source/dn001.05/paper.tex
new file mode 100644
index 0000000..176f22a
--- /dev/null
+++ b/source/dn001.05/paper.tex
@@ -0,0 +1,79 @@
+% AUTHOR: Amlal El Mahrouss
+% PURPOSE: WG05.TN001: The Mathematics of Execution.
+
+\documentclass[11pt,a4paper]{article}
+
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{hyperref}
+\usepackage[margin=1in]{geometry}
+
+\title{The Mathematics of Execution.}
+\author{Amlal El Mahrouss\\
+\texttt{amlal@nekernel.org}}
+\date{January 2026}
+
+\begin{document}
+
+\bf \maketitle
+
+\begin{center}
+	\rule[0.01cm]{17cm}{0.01cm}
+\end{center}
+
+\begin{center}
+\begin{abstract}
+This paper covers the algebraic properties and concepts of `The Execution Semantics' paper by El Mahrouss, A (2026). It is intended as an additional resource over the aforementioned paper.
+We will define their concepts alongside their properties. We assume the reader has read the previous paper on 'The Execution Semantics' by El Mahrouss, A.
+\end{abstract}
+\end{center}
+
+\begin{center}
+	\rule[1cm]{17cm}{0.01cm}
+\end{center}
+
+\section{The Execution Product}
+
+Let $\Gamma(x, y, z)$ be an execution product of variable arguments $x$, $y$, $z$ defined in compatible and composable execution context $E$: \\
+Let the index $\alpha$ denote the current execution domain of an execution product.
+\\ Let the following formula:
+\begin{equation}
+\Gamma(x, y, z) = \prod_{\alpha=1}^{n}(x_{\alpha} \times y_{\alpha} \times z_{\alpha})+C    
+\end{equation}
+ Be defined as the execution product. Where $C$ is the Unknown Execution of $\Gamma(x, y, z)$—now defined as $g(E)$.
+
+\subsection{Properties}
+
+\begin{enumerate}
+
+    \item $\Gamma$ as defined previously as the `execution product' shall always be valid within the execution context $E$.
+    \item The execution context $E$ shall not denote $\varnothing$.
+
+\end{enumerate}
+
+\section{The Unknown Execution}
+
+Let an `Unknown Execution' $U$ be defined regarding an execution product $\Gamma$. \\
+Let $N$ be defined as $\varnothing$.
+
+\subsection{Properties}
+
+\begin{enumerate}
+
+    \item $U$ shall not be equal to $N$.
+    \item The execution domain of $\Gamma$ shall not be $N$.
+
+\end{enumerate}
+
+\section{Conclusion}
+
+Such properties are essential to define `Execution Theory''s as a mathematical construct as per defined in El Mahrouss, A.' paper. 
+
+\section{References}
+
+\begin{enumerate}
+    \item El Mahrouss, A. (2026). The Execution Semantics: On Axioms, Domains, and Authority. (v2.0.0). Zenodo. https://doi.org/10.5281/zenodo.18366611
+\end{enumerate}
+
+\end{document}
diff --git a/source/dn001.06/paper.tex b/source/dn001.06/paper.tex
new file mode 100644
index 0000000..25e6c2a
--- /dev/null
+++ b/source/dn001.06/paper.tex
@@ -0,0 +1,60 @@
+
+\documentclass[11pt, a4paper]{article}
+\usepackage{graphicx}
+\usepackage{listings}
+\usepackage{xcolor}
+\usepackage{hyperref}
+\usepackage{amsfonts}
+\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
+
+\title{The $\mathbb{C}$ Index Property.}
+\author{Amlal El Mahrouss\\amlal@nekernel.org}
+\date{February 2026}
+
+\begin{document}
+\bf
+\maketitle
+\begin{center}
+	\rule[0.01cm]{17cm}{0.01cm}
+\end{center}
+\abstract{
+	This paper defines the property of $\mathbb{C}$ Indices such that $n \in \mathbb{R}$ for all $\Omega_{n} \in \mathbb{C}$. Definitions and properties are stated in this paper.
+}
+\begin{center}
+	\rule[0.01cm]{17cm}{0.01cm}
+\end{center}
+
+\section{Definitions of $\mathbb{C}$ Indices}
+
+\subsection{Definition of $\Omega$}
+
+Let $\Omega_{n}$ be a set defined as $\Omega \in \mathbb{C}$ and the following indices $\beta \in \mathbb{C}$:
+
+\begin{equation}
+	\Omega_{n} = (\beta_{0} \times \beta_{n})
+\end{equation}
+
+\subsection{Definition of $\alpha$}
+
+Let $\alpha$ denote $\Omega_{n}$ with a remainder of $\mathbb{X}$ defined as the result of $\beta_{n}$.
+\begin{equation}
+	\alpha = \Omega_{n} \cup \mathbb{X}
+\end{equation}
+
+\section{Properties of $\mathbb{C}$ Indices}
+
+\subsection{Properties of $\Omega$}
+
+\begin{enumerate}
+	\item $\Omega_{n} \cup \mathbb{X}$ such that $\mathbb{X} = \beta_{n}$.
+\end{enumerate}
+
+\subsection{Properties of $\alpha$}
+
+\begin{enumerate}
+
+	\item $\mathbb{X} = \beta_{n}$ such that $\beta_{n} \in \mathbb{C}$
+	
+\end{enumerate}
+
+\end{document}
diff --git a/source/wg04-space-rf/.keep b/source/wg04-space-rf/.keep
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/source/wg04-space-rf/.keep